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Quasimaps to quivers with potentials

Yalong Cao, Gufang Zhao

TL;DR

This work develops a rigorous framework for counting quasimaps from prestable curves to the critical locus $\mathrm{Crit}(\phi)$ of a potential on a GIT quotient, with targets arising from quivers with potentials. It combines gauged linear sigma model techniques with shifted symplectic geometry and Park's square-root virtual pullbacks to construct virtual counts and a CohFT structure, including a gluing formula that coherently assembles invariants from components. The theory yields quantum corrections to the critical cohomology $H_{F_0}(X,\varphi_{\phi})$, and provides a platform for vertex functions, Bethe equations, and connections to DT-type theories in Calabi–Yau 4-fold settings. By developing both the derived and K-theoretic perspectives and leveraging AKSZ-type constructions, the paper lays a foundation for enumerative and representation-theoretic aspects of quivers with potentials, with explicit computations in key examples like Hilb$^n(\mathbb{C}^3)$ and related quivers.

Abstract

This paper is concerned with a non-compact GIT quotient of a vector space, in the presence of an abelian group action and an equivariant regular function (potential) on the quotient. We define virtual counts of quasimaps from prestable curves to the critical locus of the potential, and prove a gluing formula in the formalism of cohomological field theories. The main examples studied in this paper is when the above setting arises from quivers with potentials, where the above construction gives quantum correction to the equivariant Chow homology of the critical locus. Following similar ideas as in quasimaps to Nakajima quiver varieties studied by the Okounkov school, we analyse vertex functions in several examples, including Hilbert schemes of points on $\mathbb{C}^3$, moduli spaces of perverse coherent systems on the resolved conifold, and a quiver which defines higher $\mathfrak{sl}_2$-spin chains. Bethe equations are calculated in these cases. The construction in the present paper is based on the theory of gauged linear sigma models as well as shifted symplectic geometry of Pantev, Toën, Vaquie and Vezzosi, and uses the virtual pullback formalism of symmetric obstruction theory of Park, which arises from the recent development of Donaldson-Thomas theory of Calabi-Yau 4-folds.

Quasimaps to quivers with potentials

TL;DR

This work develops a rigorous framework for counting quasimaps from prestable curves to the critical locus of a potential on a GIT quotient, with targets arising from quivers with potentials. It combines gauged linear sigma model techniques with shifted symplectic geometry and Park's square-root virtual pullbacks to construct virtual counts and a CohFT structure, including a gluing formula that coherently assembles invariants from components. The theory yields quantum corrections to the critical cohomology , and provides a platform for vertex functions, Bethe equations, and connections to DT-type theories in Calabi–Yau 4-fold settings. By developing both the derived and K-theoretic perspectives and leveraging AKSZ-type constructions, the paper lays a foundation for enumerative and representation-theoretic aspects of quivers with potentials, with explicit computations in key examples like Hilb and related quivers.

Abstract

This paper is concerned with a non-compact GIT quotient of a vector space, in the presence of an abelian group action and an equivariant regular function (potential) on the quotient. We define virtual counts of quasimaps from prestable curves to the critical locus of the potential, and prove a gluing formula in the formalism of cohomological field theories. The main examples studied in this paper is when the above setting arises from quivers with potentials, where the above construction gives quantum correction to the equivariant Chow homology of the critical locus. Following similar ideas as in quasimaps to Nakajima quiver varieties studied by the Okounkov school, we analyse vertex functions in several examples, including Hilbert schemes of points on , moduli spaces of perverse coherent systems on the resolved conifold, and a quiver which defines higher -spin chains. Bethe equations are calculated in these cases. The construction in the present paper is based on the theory of gauged linear sigma models as well as shifted symplectic geometry of Pantev, Toën, Vaquie and Vezzosi, and uses the virtual pullback formalism of symmetric obstruction theory of Park, which arises from the recent development of Donaldson-Thomas theory of Calabi-Yau 4-folds.
Paper Structure (66 sections, 49 theorems, 618 equations)

This paper contains 66 sections, 49 theorems, 618 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Moduli of quasimaps
  4. Virtual structures
  5. Quasimap invariants and gluing formula
  6. Quantum critical cohomology
  7. A variant of quasimaps, vertex functions, and Bethe equations
  8. Connections to other works
  9. Moduli stacks of quasimaps
  10. Set-up
  11. Stacks of twisted maps
  12. Recollection of quasimaps to GIT quotients
  13. Quasimaps to critical loci
  14. Shifted symplectic structures
  15. Derived critical locus
  16. ...and 51 more sections

Key Result

Theorem 1.1

(Theorem prop:symm_ob, Definition defi of qm vir class) The map $f$ has a canonical symmetric obstruction theory in the sense of Park Park, which is isotropic after base change via $Z(\boxplus^{n}\phi) \hookrightarrow X^n$. In particular, there is a square root virtual pullback

Theorems & Definitions (152)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 142 more